Finding Elasticity of Demand
Email: althauser@math.ucsb.edu
Instructions: You have $\mathbf{1}$ hour to complete this quiz and upload a single PDF of your work to Gradescope. You may either work in the space provided or on a separate piece of paper. Make sure to show all work and justify your solutions clearly. Please also follow the Quiz Guidelines outlined on the course Canvas page.
Problem 1
You have decided to make and sell friendship bracelets as a hobby. When bracelets are sold for $\$ 7$ each, you sell 85 bracelets each month. For every $\$ 0.50$ the price is increased you sell 2 fewer bracelets. It costs you $\$ 4$ to make each bracelet.
(a) Use calculus to determine what price bracelets should be sold at to maximize profit.
(b) How much money will you make each month at this price?
Finding the Producer Surplus
Raggs, Ltd. a clothing firm, determines that in order to sell $x$ suits, the price per suit must be $p=120-0.75 x$. It also determines that the total cost of producing $x$ suits is given by $C(x)=2000+0.75 x^{2}$
a) Find the total revenue, $R(x)$.
b) Find the total profit, $P(x)$.
c) How many suits must the company produce and sell in order to maximize profit?
d) What is the maximum profit?
e) What price per suit must be charged in order to maximize profit?
Finding the Gini Index
A small startup has five employees with annual salaries of $30,000, $40,000, $50,000, $60,000 and $70,000. Use the Gini Index to measure the income inequality among the employees.